Restricted k-color partitions, II (original) (raw)

On MultiColor Partitions with Distinct Parts

2008

denote respectively the number of m-colored partitions of n into distinct parts, distinct odd parts, and parts not divisible by r. We obtain recurrences for each of the above-mentioned types of partition functions.

New congruences modulo 5 for the number of2-color partitions

Let pk(n)be the number of 2-color partitions of nwhere one of the colors appears only in parts that are multiples of k. In this paper, we find some interesting congruences modulo 5 for pk(n)for k∈{2, 3, 4}by employing Ramanujan’s theta function identities and some identities for the Rogers–Ramanujan continued fraction. The congruence for p2(n)was earlier proved by Chen and Lin with the aid of modular forms.

Avoiding Colored Partitions of Lengths Two and Three

Pattern avoidance in the symmetric group S n has provided a number of useful connections between seemingly unrelated problems from stack-sorting to Schubert varieties. Recent work has generalized these results to S n C c , the objects of which can be viewed as "colored permutations".

The n-Color Partition Function and Some Counting Theorems

Integers, 2021

Recently, Merca and Schmidt found some decompositions for the partition function p(n) in terms of the classical Möbius function as well as Euler’s totient. In this paper, we define a counting function T r k (m) on the set of n-color partitions of m for given positive integers k, r and relate the function with the n-color partition function and other well-known arithmetic functions like the Möbius function, Liouville function, etc. and their divisor sums. Furthermore, we use a counting method of Erdös to obtain some counting theorems for n-color partitions that are analogous to those found by Andrews and Deutsch for the partition function.

Some Observations on Modulo 5 Congruences for 2-Color Partitions

2017

In this paper, we record the complete characterization of the modulo 5 congruence relation pk(25n+24−k)equiv0pmod5p_{k}(25n + 24 - k) \ \equiv \ 0 \pmod{5}pk(25n+24−k)equiv0pmod5 for kin0,1,2,ldots,24k \in \{0, 1, 2, \ldots, 24\}kin0,1,2,ldots,24, in connection with the 2-color partition function pk(n)p_k(n)pk(n), providing references to existing results for kin0,1,2,3,4,7,8,17k \in \{0, 1, 2, 3, 4, 7, 8, 17\}kin0,1,2,3,4,7,8,17, proofs for kin5,10,15,20k \in \{5, 10, 15, 20\}kin5,10,15,20 for the sake of completeness, and counter-examples in all the remaining cases. We also propose an alternative proof in the case of k=4k = 4k=4, without using the Rogers-Ramanujan ratio, thereby making the proof considerably simpler compared to the proof by Ahmed, Baruah and Ghosh Dastidar (JNT 2015).

Avoiding colored partitions of two elements in the

Enumeration of pattern-avoiding objects is an active area of study with connections to such disparate regions of mathematics as Schubert varieties and stack-sortable sequences. Recent research in this area has brought attention to colored permutations and colored set partitions. A colored partition of a set S is a partition of S with each element receiving a color from the set [k] = {1, 2, . . . , k}. Let Π n C k be the set of partitions of [n] with colors from [k].

Generalization of a few results in integer partitions

In this paper, we generalize a few important results in Integer Partitions; namely the results known as Stanley's theorem and Elder's theorem, and the congruence results proposed by Ramanujan for the partition function. We generalize the results of Stanley and Elder from a fixed integer to an array of subsequent integers, and propose an analogue of Ramanujan's congruence relations for the 'number of parts' function instead of the partition function. We also deduce the generating function for the 'number of parts', use it to provide an alternative proof of Ramaunjan's congruence relations, and relate the technical results with their graphical interpretations through a novel use of the Ferrer's diagrams.

Enumerating coloured partitions in 2 and 3 dimensions

Mathematical Proceedings of the Cambridge Philosophical Society

We study generating functions of ordinary and plane partitions coloured by the action of a finite subgroup of the corresponding special linear group. After reviewing known results for the case of ordinary partitions, we formulate a conjecture concerning a basic factorisation property of the generating function of coloured plane partitions that can be thought of as an orbifold analogue of a conjecture of Maulik et al., now a theorem, in three-dimensional Donaldson–Thomas theory. We study natural quantisations of the generating functions arising from geometry, discuss a quantised version of our conjecture, and prove a positivity result for the quantised coloured plane partition function under a geometric assumption.

A combinatorial approach to partitions with parts in the gaps

Acta Arithmetica, 1998

Many links exist between ordinary partitions and partitions with parts in the "gaps". In this paper, we explore combinatorial explanations for some of these links, along with some natural generalizations. In particular, if we let p k,m (j, n) be the number of partitions of n into j parts where each part is ≡ k (mod m), 1 ≤ k ≤ m, and we let p * k,m (j, n) be the number of partitions of n into j parts where each part is ≡ k (mod m) with parts of size k in the gaps, then p * k,m (j, n) = p k,m (j, n).

Polychromatic colorings of rectangular partitions

Discrete Mathematics, 2009

A rectangular partition is a partition of a plane rectangle into an arbitrary number of nonoverlapping rectangles such that no four rectangles share a corner. In this note, it is proven that every rectangular partition admits a vertex coloring with four colors such that every rectangle, except possibly the outer rectangle, has all four colors on its boundary. This settles a conjecture of Dinitz et al. [Y. Dinitz, M.J. Katz, R. Krakovski, Guarding rectangular partitions, in: Abstracts 23rd Euro. Workshop Comput. Geom., 2007, pp. 30-33]. The proof is short, simple and based on 4-edge-colorability of a specific class of planar graphs.